Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders

Abstract

In this work, we revisit the operator-state correspondence in the Majorana conformal field theory (CFT) with emphasis on its semion representation. Whereas the semion representation (or $Z_2$ extension of the chiral Ising CFT) gives a concise ``abelian" (or invertible) representation in the level of fusion rule and quantum states, there exists subtlety when considering the chiral multipoint correlation function. In this sense, the operator-state correspondence in the semion sector of the fermionic theory inevitably contains difficulty coming from its anomalous conformal dimension $1/16$ as a $Z_2$ symmetry operator. By analyzing the asymptotic behaviors of the existing correlation functions, we propose a nontrivial correspondence between the chiral conformal blocks and bulk correlation functions containing both order and disorder fields. One can generalize this understanding to $Z_N$ models or fractional supersymmetric models (in which there exist long-standing open problems). We expect this may improve our understanding of the simple current extension of CFT which can appear commonly in the studies of topologically ordered systems.

A figure is added, and discussions on a system with higher space-time dimensions are added (v2). Literature on recent mathematical and theoretical studies are added, and the detailed explanations of the technique are added. Typos are corrected (v3). Typos are corrected and a reference on the relation between entanglement and half-flux quantum is added (v4)